Let's analyze how to correctly find the probability of choosing a 2-character password with no 'Y's.
Step 1: Calculate the total number of possible combinations
Given the letters: Y, O, R, B, P, G, where double letters are not allowed, we can find the total number of combinations.
- The first character can be any of the 6 letters (Y, O, R, B, P, G).
- The second character can be any of the remaining 5 letters (it cannot be the same as the first).
So, the total combinations can be calculated as: \[ \text{Total combinations} = 6 \times 5 = 30 \]
Step 2: Calculate the number of combinations without 'Y'
Now, let's find the number of combinations that do not include 'Y'. The available letters are O, R, B, P, and G, which is a total of 5 letters.
- The first character can be any of the 5 letters: O, R, B, P, G.
- The second character can be any of the remaining 4 letters.
So the total combinations without 'Y' can be calculated as: \[ \text{Favorable combinations} = 5 \times 4 = 20 \]
Step 3: Calculate the probability
The probability of choosing a password with no 'Y's is given by the ratio of favorable outcomes to the total outcomes: \[ \text{Probability} = \frac{\text{Favorable combinations}}{\text{Total combinations}} = \frac{20}{30} = \frac{2}{3} \]
Conclusion
The correct probability of choosing a password with no 'Y's is \( \frac{2}{3} \).
Part 2: Identify Manuela's Error
Now let's examine the choices for Manuela's error:
A. Manuela divided the number of possible outcomes by the number of favorable outcomes. She should have divided the number of favorable outcomes by the number of possible outcomes.
B. Manuela multiplied the number of possible outcomes by the number of favorable outcomes. She should have divided the number of favorable outcomes by the number of possible outcomes.
From our calculations, we see that to find probability, the correct procedure is to divide the number of favorable outcomes by the total number of outcomes. Therefore, if Manuela ended up with an incorrect answer like three halves (which is not a valid probability), it is likely that she incorrectly divided or used the wrong numerator and denominator in her calculations.
The most fitting error from the provided options is: A. Manuela divided the number of possible outcomes by the number of favorable outcomes. She should have divided the number of favorable outcomes by the number of possible outcomes.